I keep my tubers in a stable/shed during winter in cardboard boxes and covered in hessian/fleece. Around late March/early April I pot up in potting compost and stand outside, moving them under cover if frost threatened, then back out again. Discard duff tubers that haven't survived the winter. Plant out end May in soil, perhaps a little sooner in South. I lose a few tubers each winter , but I always buy a few more to top up , chosing the colours /forms I like best that year. I am increasingly choosing single varieties with a pollen centre for thee bees , rather than the big blowsy varieties that the bees cannot dive into. (After first frost I start the process all over again by lifting them and storing in shed...I have never had the nerve to leave in soil over winter, and I think that is the right decision this year!)

Split by pulling apart by hand, no need to cut.

I don't disagree with sotongeoff, just do things differently. both methods work, it's up to you as to what suits you.